Unlike the quantum Hall bilayer, the spin within each moiré layer is still active and we have both the layer pseudospin and the real spin. In our previous work[1]}, [2]}, we consider the ideal limit where the real spin and the layer pseudospin together form a SU(4) spin. Although various interesting phases including a chiral spin liquid was predicted theoretically in the SU(4) symmetric point, we note that this ideal limit requires the interlayer distance to be almost zero, which is in tension with the requirement of zero inter-layer tunneling. Hence the realization of the ideal SU(4) model in realistic systems is a challenge. Experimentally it is much easier to completely suppress the inter-layer tunneling by putting a thick hexagon boron nitride (hBN) between the two layers[3]}, [4]}, [5]}. In Ref. gu2021dipolar,zhang2021correlated, a TMD moiré layer based on WSe\(_2\) /WS\(_2\)  on the bottom is separated from a moiré-less TMD monolayer WSe\(_2\)  on the top by a hBN barrier. Let us label the density of the two layers as \(n_t,n_b\)  per moiré unit cell. A correlated insulator with charge gap is found for the filling \(n_t=x, n_b=1-x\)  in the range \(x\in (0,x_c)\)  with the critical exciton density \(x_c\)  as large as \(0.5\)[5]}. The SU(4) symmetry is strongly broken in this case and we have only U(2)\(\times \)  U(2) symmetry corresponding to the separate conservation of charge and spin in the two layers. This moiré+monolayer setup is easier to fabricate than the moiré+moiré system which we previously proposed[1]}, [2]}. However, it is not clear whether interesting spin liquid phase can still be realized in this anisotropic case with lower symmetry. The main purpose of this work is to propose a theoretical framework to analyze the spin and exciton dynamics and demonstrate the existence of exotic phases with fractionalization even in this simpler system.
